A156861 Triangle read by rows: T(n, k) = 2^k - binomial(n, k+1).

1, 0, 2, -1, 1, 4, -2, -1, 3, 8, -3, -4, 0, 7, 16, -4, -8, -6, 3, 15, 32, -5, -13, -16, -7, 10, 31, 64, -6, -19, -31, -27, -5, 25, 63, 128, -7, -26, -52, -62, -40, 4, 56, 127, 256, -8, -34, -80, -118, -110, -52, 28, 119, 255, 512, -9, -43, -116, -202, -236, -178, -56, 83, 246, 511, 1024

Offset

0,3

Comments

Row sums are 2^n: {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...}

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n, k) = 2^k - binomial(n, k+1).

From G. C. Greubel, Nov 30 2019: (Start)

T(n, n) = 2^n.

Sum_{k=0..n-1} T(n,k) = 0. (End)

Example

Triangle begins as:
   1;
   0,   2;
  -1,   1,    4;
  -2,  -1,    3,    8;
  -3,  -4,    0,    7,   16;
  -4,  -8,   -6,    3,   15,   32;
  -5, -13,  -16,   -7,   10,   31,  64;
  -6, -19,  -31,  -27,   -5,   25,  63, 128;
  -7, -26,  -52,  -62,  -40,    4,  56, 127, 256;
  -8, -34,  -80, -118, -110,  -52,  28, 119, 255, 512;
  -9, -43, -116, -202, -236, -178, -56,  83, 246, 511, 1024;

Maple

seq(seq( 2^k - binomial(n, k+1), k=0..n), n=0..10); # G. C. Greubel, Nov 30 2019

Mathematica

Table[2^k -Binomial[n, k+1], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(PARI) T(n, k) = 2^k - binomial(n, k+1); \\ G. C. Greubel, Nov 30 2019
(Magma) [2^k - Binomial(n, k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 30 2019
(Sage) [[2^k - binomial(n, k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 30 2019
(GAP) Flat(List([0..10], n-> List([0..n], k-> 2^k - Binomial(n, k+1) ))); # G. C. Greubel, Nov 30 2019

Crossrefs

Sequence in context: A258709 A239144 A325528 * A122773 A029268 A347629

Adjacent sequences: A156858 A156859 A156860 * A156862 A156863 A156864

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 17 2009

Status

approved